Theoretical analysis on the nonlinear fractional differential equations and generalized heat equation

Using Schauder's fixed-point theorem, we establish sufficient conditions for the existence and uniqueness of solutions to the nonlinear fractional boundary value problem: \begin{cases} {}_{c}D^{\beta}\zeta(x) + f(x, \zeta(x), I^{\gamma}\zeta(x)) = 0, & x \in I = [0, 1], \quad 1 0, \\?(0) = 0, \quad \zeta(1) = \phi(\zeta), \end{cases} {(0.1)} where \phi is a functional defined on C(I, \mathbb{R}) . By constructing an appropriate Green''s function, we derive a Lyapunov-type inequality for a special case of the problem (0.1): \begin{cases} {}_{c}D^{\beta}\zeta(x) + \lambda(x)I^{\gamma}\zeta(x) = \eta(x, \zeta(x)), & x \in I = [0, 1], \quad 1 0, \\?(0) = 0, \quad \zeta(1) = \phi(\zeta). \end{cases} {(0.2)} We further make an analysis for equation (0.2) by applying the inverse operator method and the Mittag-Leffler function with illustrative examples demonstrating applications obtained. Finally, we construct an analytic solution to the following generalized fractional heat equation with an initial condition in n dimensions based on an inverse operator: \begin{cases} {}_{c}D_{t}^{\alpha}u(t, x) = \Delta_{a_{1}(x_{1}), \cdots, a_{n}(x_{n})}u(t, x) + f(t, x), & (t, x) \in \mathbb{R}^{+} \times \mathbb{R}^{n}, \quad 0 < \alpha \leq 1, \\u(0, x) = \psi(x), \end{cases} {(0.3)} where \Delta_{a_{1}(x_{1}), \cdots, a_{n}(x_{n})} = a_{1}(x_{1})\frac{\partial^{2}}{\partial x_{1}^{2}} + \cdots + a_{n}(x_{n})\frac{\partial^{2}}{\partial x_{n}^{2}}.

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